Cohomology of the Vector Fields Lie Algebra and Modules of Differential Operators on a Smooth Manifold

Let M be a smooth manifold, the space of polynomial on fibers functions on T*M (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, Vect(M), of vector fields on M with coefficients in the space of linear differential operators on . This cohomology space is closely related to the Vect(M)-modules, (M), of linear differential operators on the space of tensor densities on M of degree .     

Décroissance rapide de la distribution fλ

Let X be an open subset of ⁿ and (f₁, ...,f_p): X ^p be a holomorphic mapping. We prove that if (x⁰,0, ⁰) T^* × ^p does not belong to the characteristic variety of the _X []-module _X[]f, then there exists a conic neighborhood V × of (x⁰, ⁰) such the function is rapidely decreasing in | Im | for with Re bounded, for any (n,n)-form of class C with compact support in V. The following partial converse of this result is also established: if for all (n,n)-forms of class C with compact support in X, then .     

Intersection Cohomology on Nonrational Polytopes⋆

We consider a fan as a ringed space (with finitely many points). We develop the corresponding sheaf theory and functors, such as direct image R_* ( is a subdivision of a fan), Verdier duality, etc. The distinguished sheaf , called the minimal sheaf plays the role of an equivariant intersection cohomology complex on the corresponding toric variety (which exists if is rational). Using we define the intersection cohomology space IH(). It is conjectured that a strictly convex piecewise linear function on acts as a Lefschetz operator on IH(). We show that this conjecture implies Stanley's conjecture on the unimodality of the generalized h-vector of a convex polytope.     

Formal Dimension for Semisimple Symmetric Spaces

If G is a semisimple Lie group and (, ) an irreducible unitary representation of G with square integrable matrix coefficients, then there exists a number d() such that

On Characteristic Polynomials of Geometric Frobenius Associated to Drinfeld Modules

Let K be a function field over finite field and let be a ring consisting of elements of K regular away from a fixed place of K. Let be a Drinfeld -module defined over an -field L. In the case where L is a finite -field, we study the characteristic polynomial of the geometric Frobenius. A formula for the sign of the constant term of in terms of leading coefficient of is given. General formula to determine signs of other coefficients of is also derived. In the case where L is a global -field of generic characteristic, we apply these formulae to compute the Dirichlet density of places where the Frobenius traces have the maximal possible degree permitted by the Riemann hypothesis.     

The de Rham Bundle on a Compactification of Moduli Space of Abelian Varieties

In this paper, we show that the Chern classes c _k of the de Rham bundle defined on any good toroidal compactification of the moduli space of Abelian varieties of dimension g are zero in the rational Chow ring of , for g=4, 5 and k>0.     

On microfunctions at the boundary along CR manifolds

Let X be a complex analytic manifold, a C ² submanifold, an openset with C ² boundary .Denote by (resp. ) the microlocalization along M (resp. ) of the sheaf of holomorphic functions.In the literature (cf. [A-G], [K-S 1,2])one encounters two classical results concerning the vanishing of the cohomology groups .The most general gives the vanishing outside a range of indices j whose length is equal to (with being the number of respectively positive, negative and null eigenvalues for themicrolocal Levi form ).The sharpest result gives the concentration in a single degree, provided that the difference is locally constant for near p (with for z the base point of p).The first result was restated for the complex in [D'A-Z 2], in the case codim We extend it here to any codimension and moreover we also restate for the second vanishing theorem.We also point out that the principle of our proof, related to a criterion for constancy of sheaves due to [K-S 1], is a quite new one.     

Twists of elliptic curves

If E is an elliptic curve over , then let E(D) denote theD-quadratic twist of E. It is conjectured that there are infinitely many primesp for which E(p) has rank 0, and that there are infinitely many primes for which has positive rank. For some special curvesE we show that there is a set S of primes p with density for which if is a squarefree integer where , then E(D) has rank 0. In particular E(p) has rank 0 for every . As an example let E1 denote the curve .Then its associated set of primes S₁ consists of the prime11 and the primes p for which the order of the reduction ofX₀(11) modulo p is odd. To obtain the general result we show for primes that the rational factor of L(E(p),1) is nonzero which implies thatE(p) has rank 0. These special values are related to surjective Galois representations that are attached to modularforms. Another example of this result is given, and we conclude with someremarks regarding the existence of positive rank prime twists via polynomialidentities.     

Exceptional Sequences in Hall Algebras and Quantum Groups

Let be a finite-dimensional hereditary algebra over a finite field k, () and () be, respectively, the Hall algebra and the composition algebra of , be the isomorphism classes of finite dimensional -modules and I the isomorphism classes of simple -modules. We define and , in , to be the right and left derivations of () respectively. By using these derivations and the action of the braid group on the set of exceptional sequences of -mod, we provide an effective algorithm of calculating the root vectors of real Schur roots. This means that we get an inductive method to express u as the combinations of elements u_i in the Hall algebra, where i I and in is any exceptional -module. Because of the canonical isomorphism between the Drinfeld–Jimbo quantum group and the generic composition algebra, our algorithm is applicable directly to quantum groups. In particular, all the root vectors are obtained in this way in the finite type cases.     

A construction of representations of affine Weyl groups

Let G be a complex, semisimple, simply connected algebraic group withLie algebra . We extend scalars to the power series field in one variable C(()), and consider the space of Iwahori subalgebras containing a fixed nil-elliptic element of C(()),i.e. fixed point varieties on the full affine flag manifold. We definerepresentations of the affine Weyl group in the homology of these varieties,generalizing Kazhdan and Lusztig's topological construction of Springer'srepresentations to the affine context.     

Euler Characteristics of Local Systems on \mathcal{M}> 2

We calculate the Euler characteristics of the local systems S ^k S ² on the moduli space ₂ of curves of genus 2, where is the rank 4 local system R ¹ _* .     

Systèmes dynamiques non archimédiens et corps des norms (Non-Archimedean Dynamic Systems and Fields of Norms)

Let _k be the ring of integers of a finite extension k of the field _p of p-adic numbers. The endomorphisms of a formal group law defined over _k provide nontrivial examples of commuting formal series with coefficients in _k . This article deals with the inverse problem formulated by Jonathan Lubin within the context of non-Archimedean dynamical systems. We present a large family of series, with coefficients in _p , which satisfy Lubin's conjecture. These series are constructed with the help of Lubin–Tate formal group laws over _p . We introduce the notion of minimally ramified series which turn out to be modulo p reductions of some series of this family. The commutant monoids of these minimally ramified series are determined by using the Fontaine–Wintenberger theory of the field of norms which allows an interpretation of them as automorphisms of _p -extensions of local fields of characteristic zero. A particularly effective example illustrating the paper is given by a family of series generalizing ebyev polynomials     

On Plane Maximal Curves

The number N of rational points on an algebraic curve of genus g over a finite field satisfies the Hasse–Weil bound . A curve that attains this bound is called maximal. With and , it is known that maximalcurves have . Maximal curves with have been characterized up to isomorphism. A natural genus to be studied is and for this genus there are two non-isomorphic maximal curves known when . Here, a maximal curve with genus g ₂ and a non-singular plane model is characterized as a Fermat curve of degree .     

The Hodge Filtration and Cyclic Homology

We relate the Hodge filtration of the cohomology of a complex algebraic variety X to the Hodge decomposition of its cyclic homology. If X is smooth and projective, is the quotient of the Betti cohomology by the piece of the Hodge filtration.     

Adjoint L-Values and Primes of Congruence for Hilbert Modular Forms

Let f be a primitive Hilbert modular cusp form of arbitrary level and parallel weight k, defined over a totally real number field F. We define a finite set of primes that depends on the weight and level of f, the field F, and the torsion in the boundary cohomology groups of the Borel–Serre compactification of the underlying Hilbert-Blumenthal variety. We show that, outside , any prime that divides the algebraic part of the value at s=1 of the adjoint L-function of f is a congruence prime for f. In special cases we identify the boundary primes in terms of expressions of the form , where is a totally positive unit of F.     

Stability of Frobenius pull-backs of tangent bundles and generic injectivity of Gauss maps in positive characteristic

For a smooth projective variety X of dimension n in a projective space defined over an algebraically closed field k, the Gauss mapis a morphism from X to the Grassmannian of n-plans in sending to the embedded tangent space .The purpose of this paper is to prove the generic injectivity of Gauss mapsin positive characteristic for two cases; (1) weighted complete intersectionsof dimension of general type; (2) surfaces or 3-folds with -semistable tangent bundles; based on a criterion of Kaji by looking atthe stability of Frobenius pull-backs of their tangent bundles. The first result implies that a conjecture of Kleiman--Piene is true in case X is of general type of dimension . The second result is a generalization of the injectivity for curves.     

Fibrés d'intersection

Let f:XSbe a projective morphism of Noetherian schemes. We assume fpurely of relative dimension dand finite Tor-dimensional. We associate to d+1 invertible sheaves on Xa line bundle I _X/S ( ) on Sdepending additively on the , commuting to good base changes and which represents the integral along the fibres of fof the product of the first Chern classes of the . If d=0, I _X/S ( ) is the norm N _X/S ( ).